A206911 Position of n-th partial sum of the harmonic series when all the partial sums are jointly ranked with the set {log(k+1)}; complement of A206912.

2, 5, 8, 11, 13, 16, 19, 22, 24, 27, 30, 33, 36, 38, 41, 44, 47, 49, 52, 55, 58, 61, 63, 66, 69, 72, 74, 77, 80, 83, 86, 88, 91, 94, 97, 100, 102, 105, 108, 111, 113, 116, 119, 122, 125, 127, 130, 133, 136, 138, 141, 142, 143, 144, 145, 146, 147, 148, 149

Offset: 1

Author

Clark Kimberling, Feb 13 2012

Keywords

• nonn

Comments

Conjecture: the difference sequence of A206911 consists of 2s and 3s, and the ratio (number of 3s)/(number of 2s) tends to a number between 3.5 and 3.6.

Similar conjectures can be stated for difference sequences based on jointly ranked sets, such as A206903, A206906, A206928, A206805, A206812, and A206815.

Examples

			Let S(n)=1+1/2+1/3+...+1/n and L(n)=log(n+1).  Then
L(1)<S(1)<L(2)<L(3)<S(2)<L(4)<L(5)<S(3)<L(6)<..., so that
A206911=(2,5,8,...).
		

Crossrefs

Cf. A206912, A206815.

Programs

Mathematica

f[n_] := Sum[1/k, {k, 1, n}];  z = 300;
g[n_] := N[Log[n + 1]];
c = Table[f[n], {n, 1, z}];
s = Table[g[n], {n, 1, z}];
j = Sort[Union[c, s]];
p[n_] := Position[j, f[n]]; q[n_] := Position[j, g[n]];
Flatten[Table[p[n], {n, 1, z}]]    (* A206911 *)
Flatten[Table[q[n], {n, 1, z}]]    (* A206912 *)